Local error analysis for the Stokes equations with a singular source term

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Local error analysis for the Stokes equations with a singular source term

Loïc Lacouture

To cite this version:

Loïc Lacouture. Local error analysis for the Stokes equations with a singular source term. 2016.

�hal-01230999v2�

Local error analysis for the Stokes equations with a singular source term

Lo¨ıc LACOUTURE February 7, 2016

Abstract: The solution of the Stokes problem with a punctual force in source term is not H¹ˆL² and therefore the approximation by a finite element method is suboptimal. In the case of Poisson problem with a Dirac mass in the right-hand side, an optimal convergence for the Lagrange finite elements has been shown on a subdomain which excludes the singularity in L²-norm by K¨oppl and Wohlmuth, and, independently, Bertoluzza and co-authors have proved a quasi-optimal convergence in H^s-norm, for s ě1. Here we show a quasi-optimal local convergence in H¹ˆL²- norm for a Pk{Pk´1-finite element method, k ě2, and for the P1b{P¹. The error is still analysed on a subdomain which does not contain the singularity. The proof is based on local Arnold and Liu error estimates, a weak version of Aubin-Nitsche duality lemma applied to the Stokes problem and discrete inf-sup conditions. These theoretical results are generalized to a wide class of finite element methods, before being illustrated by numerical simulations.

Key words: Stokes problem, Dirac measure, Stokeslet, finite element method, local error estimates.

1 Introduction.

This paper is about the accuracy of the finite element method to solve the Stokes problem with a punctual force in source term. Let us consider this following problem

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´△u`∇p “ δ^x0F in Ω, divpuq “ 0 in Ω, u “ 0 on BΩ,

(1)

where Ω Ă R² is a square, and δx₀F denotes the punctual force F located at x₀ P Ω such that distpx0,Ωq ą0.

Our interest in Problem (1) is motivated by the modeling of the movement of thin structures in a viscous fluid, such as flagella connected to bacteria or cilia involved in the muco-ciliary transport in the lung [12]. Indeed, for instance in the lung, the cilium is very thin and a direct simulation with a graded mesh would be too expensive. In the asymptotics of a zero diameter cilium and an infinite velocity, the cilium is thus replaced by a line Dirac of forces in source term. In order to ease the computations, the line Dirac of forces is approached by a sum of punctual forces distributed along the cilium [16]. Finally, by linearity of the Stokes problem, the analysis of the subsequent problem reduces to Problem (1).

In dimension 2, Problem (1) has no H¹pΩq²ˆL²pΩq-solution. Although the numerical solution can be defined in this case, theH¹pΩq-error (respectivelyL²pΩq-error) for the velocity (respectively

the pressure) has no sense, and theL²-estimates of the velocity cannot be derived like in the regular case without suitable modifications.

Let us notice that the scalar version of this problem, the Poisson Problem with a Dirac mass in right-hand side, has already been widely studied. It occurs in many applications from different areas like in optimal control of elliptic problems with state contraints [8] or in the mathematical modeling of electromagnetic fields [14]. Problems of this type are found in controllability for elliptic parabolic equations [9,10,17] and in parameter identification problems with pointwise measurements [19]. In this case, Babu˜ska proved in [3] for two-dimension smooth domain an L²pΩq-convergence of order h^1´ε, εą 0, where h is the mesh size, and Scott has shown in [20] an a priori error estimates of order 2´^d2, where the dimensiondis 2 or 3. Casas has got the same result in [7] for general Borel measures on the right-hand side.

To the best of our knowledge, there is no finite element method convergence result for the Stokes problem with a punctual force in source term. Moreover, in applications, the punctual force (or the Dirac measure) at the pointx₀ is often a model reduction approach and the finite element solution does not need to approximate precisely the exact solution at the point x0. Thus, it is interesting to estimate the error on a fixed subdomain which does not contain the singularity. In the case of the Poisson problem, K¨oppl and Wohlmuth recently showed in [15] a quasi-optimal local convergence for low order inL²-norm for Lagrange finite elements and optimal local convergence for higher orders.

A quasi-optimal local convergence inH^s-norm, sě1 and an optimal local convergence in the case of low order have been proved in dimension 2 in [5]. In this paper, we establish still in dimension 2 local error estimates for the Stokes problem with a punctual force in source term, Problem (1), and show a quasi-optimal convergence in H¹ˆL²-norm. The proof is based on the Arnold and Liu Theorem [2] that establishes local error estimates for finite element discretizations of the Stokes equations with regular source term. It is written in the case of thePk{Pk´1 elements forkě2, and the MINI finite element method P1b{P¹ if k“1. No graded meshes are required for these results and they imply that there is no pollution effects far from the singularity.

The paper is organized as follows. Our main result is formulated in Section 2 followed by the Arnold and Liu Theorem [2], an important tool for the proof presented in Section 3. Our theo- retical results are generalized in Section 4, before being illustrated in Section5 by some numerical simulations.

2 Main results.

In this section, we first set all the notations used in this paper. Then, we formulate our main result and give an important tool for the proof: the Arnold and Liu Theorem. For the sake of clarity, this result is first set and proved in the particular case of the Pk{Pk´1 finite elements (kě2) and the P1b{P¹ elements. It will be generalized in Section 4.

2.1 Notations.

For a domainD, we will denote by} ¨ }^s,q,D(respectively| ¨ |^s,q,D) the norm (respectively semi-norm) of the Sobolev spaceW^s,qpDq, and by}¨}^s,D (respectively|¨|^s,D) the norm (respectively semi-norm) of the Sobolev space H^spΩq. Letters in bold refer to a vector ofR² or a function with values inR². For the numerical solution, let us introduce a family of quasi-uniform simplical triangulationsT_h of Ω, where h is the meshsize. For the approximation spaces V_h^k and W_h^k, we will use thePk{Pk´1

x₀ Ω

Ω0

Ω1

BΩ0 BΩ1

mesh

Figure 1: Domains Ω0 and Ω1. finite elements, for kě2, defined as

V_h^k “ v_h PCpΩ¯q²ˇ

ˇv_h|T PP_krTs,@T PT_h( , W_h^k “!

p_h PCpΩ¯qˇ

ˇˇp_h|T PP_k´1rTs,@T PT_h )

,

and if k“1, we will use the MINI finite element methodP1b{P¹, whereP1b denotes the continous piecewise linear and bubble functions. For a finite element T, the bubble function b is defined by

bpxq “

# λ^T₁pxqλ^T₂pxqλ^T₃pxq if xPT,

0 else,

where λ^T₁,λ^T₂ and λ^T₃ are the barycentric coordinates ofx in relation to the triangle T.

We fix two subdomains of Ω, called Ω0 and Ω1, such that Ω0 ĂĂ Ω1 ĂĂΩ and x₀ R Ω1 (see Figure 1). We consider a mesh which satisfies the following condition:

Assumption 1. For some h₀, we have for all 0ăhďh₀ (see Figure 1), Ω^m₀ ^ŞΩ^c₁“ H, where Ω^m₀ “ ď

TPT_h T^ŞΩ0‰H

T.

2.2 Statement of our main results.

Our main result is given by Theorem 1. The rest of the paper is mostly concerned by the proof, the generalization and the illustration of this theorem.

Theorem 1. Consider Ω0 ĂĂΩ1 ĂĂ Ω satisfying Assumption 1, k ě1, 1 ď q ă 2, let pu, pq P W₀^1,qpΩq ˆL^q₀pΩq be the solution of Problem (1) andpu_h, p_hq its Galerkin projection ontoV_h^kˆW_h^k satisfying ş

Ωp_h “0 and ż

Ω

∇pu´uhq::∇η´ ż

Ωpp´phqdivpηq “ 0 for all ηPV_h^k, ż

Ω

divpu´u_hqξ “ 0 for all ξPW_h^k.

(2)

Under the assumption that pu, pq PH^k`1pΩ1q²ˆH^kpΩ1q, there exists h1 such that if 0 ăh ďh1, we have,

}u´uh}^1,Ω0` }p´ph}^0,Ω0 ďCpΩ0,Ω1,Ωqh^ka

|lnh|.

2.3 Regularity of the solution pu, pq.

In this subsection, we focus on the singularity of the solution, which is the main difficulty in the study of this kind of problems. In dimension 2, Problem (1) has a unique weak solution pu, pq PW^1,qpΩq²ˆL^q0pΩq for all q P r1,2r. Indeed, the 2d Stokeslet denoting pu_δ, p_δq is defined as (see for instance [18])

uδpxq “ 1 4π

ˆ

´ln}x}I2`x¨^tx }x}²

˙ F, p_δpxq “ x¨F

2π}x}².

(3)

The Stokeslet pu_δ, p_δq satisfies inD¹pR²q

"

´△uδ`∇pδ “δ0F, divpu_δq “0,

so that the Stokeslet pu_δp¨ ´x0q, p_δp¨ ´x0qq contains the singular part of pu, pq, the solution of Problem (1). As it is done in [1] in the case of the Poisson problem, the solution pu, pq can be built by using a suitable lift procedure which consists in adding to puδ, pδq a corrector term pw, πq PH¹pΩq²ˆL²0pΩq, which satisfies the following problem:

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´△w`∇π “ 0 in Ω,

divpwq “ 0 in Ω,

w “ ´uδp¨ ´x0q onBΩ.

Then, the solution is given by:

upxq “uδpxq `wpxq “ 1 4π

ˆ

´ln}x}I2`x¨^tx }x}²

˙

F`wpxq, ppxq “p<sub>δ</sub>pxq `πpxq “ x¨F

2π}x}² `πpxq.

Moreover, it is easy to show that uδ R H₀¹pΩq² and p_δ R L²pΩq. Actually, we can specify how the quantity |uδ|^1,q,Ω goes to infinity when q goes to 2, with q ă 2 (which will be noted q Ñ_ă 2).

According to (3), estimating|u_δ|1,q,Ω whenq Ñ

ă 2 is reduced to estimate|u_δ|1,q,B, whereB denotes the ball Bpx0,1q: we can easily show that there exists Cą0 depending only onF such that

@1ďq ă `8, u_δ PL^qpΩq and |∇u_δ| ď C }x}, and so, using polar coordinates, we get for qă2,

|uδ|^q1,q,Ω“ ż

B|∇u_δpxq|^qdxď ż

B

C^q

}x}^qdx“C^q ż1

0

ż2π

0

1

r^q´1dθdr“2πC^q 1 2´q. Finally, there exists C ą0 independent of q such that, for 1ďq ă2,

|uδ|^1,q,Ω ď C

?2´q. (4)

In the same way, we can easily show that there exists C ą 0 independent of q such that, for 1ďq ă2,

|p_δ|0,q,Ω ď C

?2´q. (5)

2.4 Arnold and Liu Theorem.

Before stating Arnold and Liu Theorem, let us enumerate the assumptions that the finite element spacesV_h^k and W_h^k have to satisfy so that the theorem is true.

Assumption 2. Given two fixed concentric spheres B0 and B with B0 ĂĂB ĂĂΩ, there exists an h0 such that for all 0ăhďh0, we have for some integers k1 and k2:

B1 For any 1ďℓ, for each vPH^ℓpBq², there existsηPV_h^k such that }v´η}1,B ďCh^r1´1}v}ℓ,B, r₁ “minpk₁`1, ℓq. For any 0ďs, for each πPH^spBq, there existsξ PW_h^k such that

}π´ξ}0,B ďCh^r2}π}s,B, r₂“minpk₂`1, sq.

Moreover, if v P H₀¹pB₀q² (respectively π vanishes on BzB₀) then η (respectively ξ) can be chosen to satisfy η PH₀¹pBq² (respectively ξ vanishes on ΩzB).

B2 Let ϕ P C₀⁸pB0q, vh P V_h^k and πh P W_h^k, then there exist η P V_h^kŞH₀¹pBq and ξ P W_h^k with supp ξĂB such that

}ϕv_h´η}^1,B ďCpϕ, B, B₀qh}v_h}^1,B, }ϕπ_h´ξ}0,B ďCpϕ, B, B₀qh}π_h}0,B.

B3 For each 0ăhďh₀ there exists a domain B_h withB₀ĂĂB_hĂĂB such that for any 0ďℓ, for all vhPV_h^k and π_hPW_h^k, we have

}v_h}^1,BhďCh^´1´ℓ}v_h}´ℓ,Bh, }π_h}^0,Bh ďCh^´ℓ}π_h}´ℓ,Bh.

B4 There exists βą0such that for all 0ăhďh0, there is a domain Bh, with B0 ĂĂBhĂĂB for which

inf

π_hPW^k

h

suppπ_hĂBh

sup

v_hPV^k

h

suppv

hĂBh

ş

Bhdivpv_hqπ_h

|π_h|0,Bh|v_h|1,Bh

ěβ ą0.

We now state the following theorem by Arnold and Liu [2], a key tool in the forthcoming proof of Theorem 1.

Theorem (Arnold and Liu [2]). Consider Ω0 ĂĂ Ω1 ĂĂ Ω, V_h^k and W_h^k satisfy Assumption 2.

Suppose that pv, πq P H¹pΩq² ˆL²pΩq satisfies pv, πq|_Ω1 P H^ℓpΩ1q² ˆH^ℓ´1pΩ1q for some ℓ ą 0.

Suppose that pv_h, π_hq PV_h^kˆW_h^k satisfies ş

Ωπ´π_h “0 and ż

Ω

∇pv´vhq::∇η´ ż

Ωpπ´πhqdivpηq “ 0 for all ηPV_h^k, ż

Ω

divpv´v_hqξ “ 0 for all ξPW_h^k.

Let t be a nonnegative integer. Then there exist a constant C ą0 and a realh1 ą0 depending only on Ω1,Ω0, and t, such that if0ăhďh₁ we have

}v´v_h}1,Ω0` }π´π<sub>h</sub>}0,Ω0 ďCph<sup>r1´1</sup>}v}ℓ,Ω1 `h^r2´1}π}ℓ´1,Ω1

` }v´v<sub>h</sub>}´t,Ω1` }π´π_h}´t´1,Ω1q, where r1 “minpk1`1, ℓq, r2 “minpk2`2, ℓq, and k1, k2 as in Assumption B1.

AssumptionsB1andB3are quite standard and satisfied by a wide class of finite element spaces, including all finite element spaces defined on quasi-uniform meshes [11]. The parameters k1 and k2

play respectively the role of the order of approximation of the spaces V_h^k andW_h^k. In our paper, for kě2, we will havek₁ “kandk₂ “k´1, and fork“1, we will havek₁ “k₂ “k“1. Assumption B2is less common but also satisfied by a wide variety of approximation spaces, including the P1b- finite elements [2]. Actually, for Lagrange finite elements, a stronger property than assumptionB2 is shown in [4]: let 0ďsďℓďk,ϕPC⁸

0 pBq and v_h PV_h^k, then there exists ηPV_h^k such that }ϕv_h´η}s,B ďCpϕqh^ℓ´s`1}v_h}ℓ,B. (6) Applied for s“ℓ“1, inequality (6) gives assumptionB2. When B_h “Ω, Assumption B4is the standard stability condition or discrete inf-sup condition of the Stokes elements. It usually holds as long as B_h is a union of elements.

Remark 1. The assumptionpv, πq PH¹pΩq²ˆL²pΩqis not necessary, but it ensures that the finite element solution pvh, πhq is well-defined. In the Dirac right-hand side case, as V_h^k Ă CpΩq, the discrete solution pu_h, p_hq is well-defined and Arnold and Liu Theorem holds.

3 Proof of Theorem 1.

This section is devoted to the proof of our main result: Theorem1. First, we show a weak version of Aubin-Nitsche duality lemma (Lemma1), then we establish two discrete inf-sup conditions (Lemmas 2 and 3), and finally we use these results to prove Theorem1.

3.1 Aubin-Nitsche duality lemma with a singular source term.

The proof of Theorem 1 is based on Arnold and Liu Theorem. In order to estimate the quantities }u´uh}´t,Ω1 and }p´ph}´t´1,Ω1, we will first show a weak version of Aubin-Nitsche Lemma in the case of the Stokes Problem with a singular source term.

Lemma 1. Consider f PW^´1,qpΩq² “ pW₀^1,q1pΩq²q¹, 1ăq ă2, and let pw, πq P W₀^1,qpΩq ˆL^qpΩq be the unique solution of $

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´△w`∇π “ f in Ω, divpwq “ 0 in Ω, w “ 0 on BΩ.

Let pwh, πhq be the Galerkin projection of pw, πq in V_h^kˆW_h^k. For any integer 0ďtďk´1, }w´wh}^´t,Ω` }π´π_h}^´t´1,Ω

ďCh^2p1{q1´1{2qh<sup>t`1</sup>p|w´w<sub>h</sub>|1,q,Ω` |π´π_h|0,q,Ωq. (7)

Proof. We aim at estimating, fortě0, the H^´tpΩq-norm and theH^´t´1pΩq-norm respectively of the errors w´w_h and π´π_h:

}w´w_h}´t,Ω“ sup

ϕPC₀⁸pΩq²

1 }ϕ}t,Ω

ˇˇ ˇˇ ż

Ωpw´w_hq ¨ϕ ˇˇ

ˇˇ (8)

}π´π_h}´t´1,Ω “ sup

ψPC₀⁸pΩq

1 }ψ}t`1,Ω

ˇˇ ˇˇ ż

Ωpπ´π_hqψ ˇˇ

ˇˇ (9)

The Galerkin projection pw_h, π_hq satisfies ż

Ω

π´π_h“0 and ż

Ω

∇pw´w_hq::∇η´ ż

Ωpπ´π_hqdivpηq “ 0 for all ηPV_h^k, ż

Ω

divpw´whqξ “ 0 for all ξPW_h^k.

(10)

Consider ϕPC⁸

0 pΩq² and letpw^ϕ, π^ϕq PH^{t`2</sup>pΩq ˆH<sup>t`1}pΩq be the solution of

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´△w^ϕ`∇π^ϕ “ ϕ in Ω, divpw^ϕq “ 0 in Ω, w^ϕ “ 0 onBΩ.

Existence and uniqueness of the solution to this problem are given in [21] (see Chapter I, §2), and we have the estimate

}w^ϕ}t`2,Ω` }π^ϕ}t`1,ΩďC}ϕ}t,Ω. (11)

In dimension 2, by the Sobolev injections established for instance in [6], we have H^t`2pΩq ĂW^1,q1pΩq,

H^t`1pΩq ĂL^q1pΩq, (12)

for all q¹ inr2,`8r. Thus ż

Ωpw´whq ¨ϕ“ ´ ż

Ωpw´whq ¨△w^ϕ` ż

Ωpw´whq ¨∇π^ϕ

“ ż

Ω

∇pw´w_hq::∇w^ϕ´ ż

Ω

divpw´w_hqπ^ϕ. By adding (10) in the last equality, we get for anyη PV_h^k and any ξPW_h^k,

ż

Ωpw´w_hq ¨ϕ“ ż

Ω

∇pw´w_hq::∇pw^ϕ´ηq ´ ż

Ω

divpw´w_hqpπ^ϕ´ξq

` ż

Ω

divpηqpπ´πhq. By definition of w^ϕ, divpw^ϕq “0 on Ω, so

ż

Ωpw´w_hq ¨ϕ“ ż

Ω

∇pw´w_hq::∇pw^ϕ´ηq ´ ż

Ω

divpw´w_hqpπ^ϕ´ξq

` ż

Ω

divpη´w^ϕqpπ´π_hq ď|w´w_h|1,q,Ω

`|w<sup>ϕ</sup>´η|1,q<sup>1</sup>,Ω` |π^ϕ´ξ|0,q¹,Ω

˘

` |π´π_h|^0,q,Ω|w^ϕ´η|^1,q1,Ω.

Now let us deal with the pressure estimate. For any ψPC₀⁸pΩq, we denote byψrthe function ψr“ψ´ 1

|Ω| ż

Ω

ψ.

By definition, it is easy to see that ψrsatisfies ż

Ω

ψr“0 and @tě0, }ψr}t`1,Ω ďCpΩq}ψ}t`1,Ω.

We can now establish the result for the pressure: considerψPC₀⁸pΩqand letpw^ψ, π^ψq PH^{t`2</sup>pΩqˆ H<sup>t`1}pΩq be the solution of

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´△w^ψ`∇π^ψ “ 0 in Ω, divpw^ψq “ ψr in Ω, w^ψ “ 0 onBΩ,

See [21] (Chapter I,§2) for the existence and the uniqueness, and the following estimate

}w^ψ}t`2,Ω` }π^ψ}t`1,Ω ďC}ψr}t`1,Ω ďC}ψ}t`1,Ω. (13) Moreover,

ż

Ω

π´π_h“0, so that ż

Ωpπ´π_hqψ“ ż

Ωpπ´π_hqψr` 1

|Ω| ż

Ω

ψ ż

Ω

π´π_h “ ż

Ωpπ´π_hqψ.r

By the Sobolev injections recalled in (12), and the Galerkin projection property (10), we can write for all ηPV_h^k,

ż

Ωpπ´πhqψ“ ż

Ωpπ´πhqψr

“ ż

Ωpπ´π_hqdivpw^ψq

“ ż

Ωpπ´π_hqdivpw^ψ´ηq ` ż

Ω

∇pw´w_hq::∇η.

Then, for all vPW₀^1,qpΩq ż

Ω

∇w^ψ ::∇v´ ż

Ω

π^ψdivpvq “0, so, with v“w´w_h, and for any ξPW_h^k,

ż

Ωpπ´π_hqψ“ ż

Ωpπ´π_hqdivpw^ψ´ηq ` ż

Ω

∇pw´w_hq::∇pη´w^ψq

` ż

Ω

π^ψdivpw´w_hq

“ ż

Ωpπ´π_hqdivpw^ψ´ηq ` ż

Ω

∇pw´w_hq::∇pη´w^ψq

` ż

Ωpπ^ψ´ξqdivpw´w_hq ď|π´π_h|^0,q,Ω|w^ψ´η|^1,q1,Ω

` |w´wh|^1,q,Ω´

|w^ψ ´η|1,q¹,Ω` |π^ψ´ξ|0,q¹,Ω

¯ .

Finally, for any pη1, ξ1q PV_h^kˆW_h^k, ż

Ωpw´w_hq ¨ϕď|w´w_h|1,q,Ω

`|w<sup>ϕ</sup>´η<sub>1</sub>|1,q<sup>1</sup>,Ω` |π^ϕ´ξ₁|0,q¹,Ω

˘

` |π´πh|^0,q,Ω|w^ϕ´η1|^1,q1,Ω, (14) and for any pη2, ξ2q PV_h^kˆW_h^k,

ż

Ωpπ´π_hqψď|π´π_h|0,q,Ω|w^ψ´η₂|1,q¹,Ω

` |w´w_h|1,q,Ω

´|w^ψ´η₂|1,q¹,Ω` |π^ψ´ξ₂|0,q¹,Ω

¯

. (15)

In order to estimate|w^ϕ´η₁|1,q¹,Ω,|w^ψ´η₂|1,q¹,Ω,|π^ϕ´ξ₁|0,q¹,Ω and|π^ψ´ξ₂|0,q¹,Ω, we will need the following result:

Proposition 1 (Girault, Raviart, Corollary A.2, page 97 [13]). Let T_h be a family of quasi-uniform simplicial triangulations of ΩĂR², where h is the meshsize. For any 0ďm ďt`1 ďk, for any mesh element T in the family, for any vPW<sup>k`1,q1</sup>pΩq, q¹ě2 real,

|v´Π_hv|m,q¹,T ďCh^2p1{q1´1{2qh<sup>t`2´m</sup>|v|t`2,2,T, (16) where Π_hv is the P_k-interpolant of the function v.

Up to now and until the end of this proof, we will take

η₁ “Π_hw^ϕ and η₂ “Π_hw^ψ PV_h^k, ξ1 “Πr_hπ^ϕ and ξ2 “Πr_hπ^ψ PW_h^k,

where Π_hv is the P_k-interpolant of the functionv and Πr_hv is the P_k´1-interpolant of the function v. By (16), with m“1, 0ďtďk´1, for all T finite element in the family,

|w^ϕ´η₁|1,q¹,T ďCh^2p1{q1´1{2qh<sup>t`1</sup>|w<sup>ϕ</sup>|t`2,2,T, (17)

|w^ψ´η₂|1,q¹,T ďCh^2p1{q1´1{2qh<sup>t`1</sup>|w<sup>ψ</sup>|t`2,2,T, and with m“0,

|π^ϕ´ξ1|^0,q1,T ďCh^2p1{q1´1{2qh<sup>t`1</sup>|π<sup>ϕ</sup>|t`1,2,T,

|π^ψ´ξ2|0,q¹,T ďCh^2p1{q1´1{2qh<sup>t`1</sup>|π<sup>ψ</sup>|t`1,2,T. We denote the triangles of the mesh bytT_iu^{i“1,¨¨¨,N}, and we set

a“ pa_iqi andb“ pb_iqi, wherea_i“ |w^ϕ´η₁|1,q¹,Ti andb_i “ |w^ϕ|t`2,2,Ti. By (17), we have, for alliin rr1, Nss,

ai ďCh^2p1{q1´1{2qh^t`1bi. We recall the norm equivalence in R^N for 0ărăs,

}x}^ℓs ď }x}^ℓr ďN^1{r´1{s}x}^ℓs,

with here N „Ch^´2. As 2ăq¹, we have }b}ℓ^q1 ď }b}ℓ². Then, we can write

|w^ϕ´η₁|1,q¹,Ω “ }a}_ℓ^q1 ďCh^{t`1</sup>h<sup>2p1{q1´1{2q</sup>}b}<sub>ℓ</sub><sup>q1</sup> ďCh<sup>t`1}h^2p1{q1´1{2q}b}ℓ²

ďCh^{t`1</sup>h<sup>2p1{q1´1{2q</sup>|w<sup>ϕ</sup>|<sup>t`2,2,Ω}. Similarly, we get

|w^ψ´η₂|1,q¹,Ω ďCh<sup>t`1</sup>h<sup>2p1{q1´1{2q</sup>|w<sup>ψ</sup>|t`2,2,Ω,

|π^ϕ´ξ₁|0,q¹,Ω ďCh<sup>t`1</sup>h<sup>2p1{q1´1{2q</sup>|π<sup>ϕ</sup>|t`1,2,Ω,

|π^ψ´ξ₂|0,q¹,Ω ďCh<sup>t`1</sup>h<sup>2p1{q1´1{2q</sup>|π<sup>ψ</sup>|t`1,2,Ω, and by (11) and (13), we get

|w^ϕ´η1|^1,q1,Ω ďCh^t`1h^2p1{q1´1{2q}ϕ}^t,2,Ω,

|w^ψ ´η2|1,q¹,Ω ďCh<sup>t`1</sup>h<sup>2p1{q1´1{2q</sup>}ψ}t`1,2,Ω,

|π^ϕ´ξ1|0,q¹,Ω ďCh^t`1h^2p1{q1´1{2q}ϕ}^t,2,Ω,

|π^ψ ´ξ₂|0,q¹,Ω ďCh<sup>t`1</sup>h<sup>2p1{q1´1{2q</sup>}ψ}t`1,2,Ω,

Finally, the proof is ended by combining (8), (9), (14), (15), and the last inequalities.

Corollary 1. Letpu_h, p_hq PV_h^kˆW_h^kbe the Galerkin projection of the solutionpu, pqof Problem (1), for any 0ăεă1,

}u´uh}´k`1,Ω` }p´ph}´k,Ω

ďCh^´εh^kp|u´u_h|1,qε,Ω` |p´p_h|0,qε,Ωq, whereq_εP r1,2ris defined by

qε“ 2 1`ε

ˆ

and soq_ε¹ “ 2 1´ε

˙

. (18)

Proof. We will apply Lemma 1 with F “ δx₀F, w “ u, π “ p and t “ k´1. We can explicit inequality (7):

2 ˆ1

q¹_ε ´1 2

˙

“2

ˆ1´ε 2 ´1

2

˙

“ ´ε, (19)

and so, it follows

}u´u_h}´k`1,Ω` }p´p_h}´k,Ω

ďCh^´εh^kp|u´uh|^1,qε,Ω` |p´ph|^0,qε,Ωq.

3.2 Discrete inf-sup conditions in L^qε-norm.

Section 3.3 is devoted to estimate of |u´uh|^1,qε,Ω and |p´ph|^0,qε,Ω. In that prospect, we need to establish two discrete inf-sup conditions.

Lemma 2. With q_ε and q_ε¹ defined in (18), the approximation space ˚V_h^k defined by

˚V_h^k“

"

vhPV_h^k ˇˇ ˇˇ ż

Ω

divpvhqph “0,@phPW_h^k

* , satisfies the following discrete inf-sup condition:

inf

u_hP˚V_h^k

sup

vhP˚V_h^k

ş

Ω∇uh::∇vh

|u_h|1,qε,Ω|v_h|1,q¹_ε,Ω ěCh^ε. Proof. The bilinear form

apu,vq “ ż

Ω

∇u::∇v

is continuous and coercive on H₀¹pΩq, so for ˚V_h^k vector subspace of H₀¹pΩq, we have the inf-sup condition:

inf

uhP˚V_h^k

sup

vhP˚V_h^k

ş

Ω∇u_h ::∇v_h

|u_h|^1,Ω|v_h|^1,Ω ěαą0, where α only depends on Ω. We recall the following inverse inequality:

Proposition 2 (Ciarlet, Theorem 3.2.6, page 140 [11]). Let Th a family of quasi-uniform simplicial triangulations of ΩĂR^d, where h is the meshsize. For v_hPV_h^k, 1ďr, să `8, 0ďℓďm,

˜ ÿ

TPT_h

|v_h|^rm,r,T

¸1{r

ďCh´drmaxt0,1{s´1{rush^´pm´ℓq

˜ÿ

TPT_h

|v_h|^sℓ,s,T

¸1{s

.

We will apply this to any v_h PV˚_h^kĂCpΩq, with d“2,m“l“1,s“2 andr “q_ε¹ to get:

|vh|^1,qε¹,ΩďCh^{´2p1{2´1{q1εq}|vh|^1,2,Ω “Ch^´ε|vh|^1,2,Ω, using equality (19). Moreover, for any u_h PV˚_h^k,

|u_h|1,qε,ΩďC|u_h|1,2,ΩďC sup

vhP˚V_h^k

ş

Ω∇u_h::∇v_h

|v_h|1,2,Ω ďCh^´ε sup

vhPV˚_h^k

ş

Ω∇u_h ::∇v_h

|v_h|1,q¹_ε,Ω

Finally,

inf

u_hP˚V_h^k

sup

vhP˚V_h^k

ş

Ω∇uh::∇vh

|u_h|1,qε,Ω|v_h|1,q¹_ε,Ω ěCh^ε.

The second discrete inf-sup condition we need is given by the following lemma:

Lemma 3. With q_ε and q_ε¹ defined in (18), the approximations spaces V_h^k and W_h^k satisfy the following discrete inf-sup condition:

inf

p_hPW_h^k

sup

v_hPV_h^k

ş

Ωdivpv_hqp_h

|p_h|^0,qε,Ω|vh|^1,q1ε,Ω ěCh^ε.

Proof. The proof is similar to the proof of Lemma2. According to Assumption B4, inf

p_hPW_h^k

sup

vhPV_h^k

ş

Ωdivpv_hqp_h

|p_h|0,Ω|v_h|1,Ω ěβ ą0.

According to Proposition 2, for any vh PV_h^k,

|v_h|1,q¹_ε,Ω ďCh^´ε|v_h|1,2,Ω. So, we have, for any p_h PW_h^k and q_εă2,

|p_h|0,qε,Ω ďC|p_h|0,Ω ďC sup

v_hPV_h^k

ş

Ωdivpv_hqp_h

|v_h|^1,2,Ω ďCh^´ε sup

v_hPV_h^k

ş

Ωdivpv_hqp_h

|v_h|^1,q1ε,Ω

Finally, we get

inf

p_hPW_h^k sup

v_hPV_h^k

ş

Ωdivpv_hqp_h

|p_h|^0,qε,Ω|v_h|^1,q1ε,Ω ěCh^ε.

3.3 Estimates of |u´u_h|1,qε,Ω and |p´p_h|0,qε,Ω.

Following Corollary 1, the quantities |u´u_h|^1,qε,Ω and|p´p_h|^0,qε,Ω have to be estimated to prove Theorem 1. We will apply the last two results to bound them in terms of|u|1,qε,Ω and |p|0,qε,Ω. Lemma 4. Let pu_h, p_hq PV_h^kˆW_h^kbe the Galerkin projection of the solution pu, pqof Problem (1), for any small enough real εą0,

|u´uh|^1,qε,ΩďCh^´εp|u|^1,qε,Ω` |p|^0,qε,Ωq.

Proof. First, we will estimate |u_h|^1,qε,Ω in terms of|u|^1,qε,Ω. As we have divpuq “0 on Ω, by (2) we

have ż

Ω

divpu_hqq_h “0, @q_hPW_h^k,

and so, u_h P˚V_h^k. According to Lemma2, there exists v_h P˚V_h^k such as |v_h|^1,q1ε,Ω “1, and

|u_h|^1,qε,Ω ďCh^´ε ż

Ω

∇u_h ::∇v_h. Moreover, equality (2) gives

ż

Ω

∇u_h ::∇v_h “ ż

Ω

∇u::∇v_h´ ż

Ω

divpv_hqpp´p_hq.

Now, v_h P˚V_h^k, so ż

Ω

divpv_hqp_h“0.

Finally, as |v_h|1,q¹_ε,Ω “1, we get

|u_h|1,qε,ΩďCh^´ε ˆż

Ω

∇u::∇v_h´ ż

Ω

divpv_hqp

˙ , ďCh^´εp|u|^1,qε,Ω` |p|^0,qε,Ωq.

We conclude with the triangulary inequality,

|u´u_h|^1,qε,Ωď |u|^1,qε,Ω` |u<sub>h</sub>|<sup>1,q</sup>ε,Ω ďCh<sup>´ε</sup>p|u|<sup>1,q</sup>ε,Ω` |p|^0,qε,Ωq.

We can now estimate|p´p_h|^0,qε,Ω.

Lemma 5. Let puh, phq PV_h^kˆW_h^kbe the Galerkin projection of the solution pu, pqof Problem (1), for any small enough real εą0,

|p´p_h|^1,qε,Ω ďCh^´2εp|u|^1,qε,Ω` |p|^0,qε,Ωq.

Proof. The proof is similar to the velocity case: according to Lemma 3, there exists vh PV_h^k such as|v_h|1,q¹_ε,Ω“1 and

|p_h|^0,qε,Ω ďCh^´ε ż

Ω

divpvhqp_h. By (2), we have ż

Ω

divpvhqph “ ´ ż

Ω

∇pu´uhq::∇v_h` ż

Ω

divpvhqp.

By applying Lemma 4, as |v_h|^1,qε¹,Ω“1, we get

|ph|^0,qε,ΩďCh^´ε ˆ

´ ż

Ω

∇pu´uhq::∇vh` ż

Ω

divpvhqp

˙ , ďCh^´εp|u´u_h|1,qε,Ω` |p|0,qε,Ωq,

ďCh^´2εp|u|^1,qε,Ω` |p|^0,qε,Ωq.

3.4 Proof of Theorem 1.

We can now prove Theorem 1.

Proof. The functionsuandpare analytic on Ω1, so the quantities}u}k`1,Ω1 and}p}^k,Ω1are bounded.

Let us note that in this case pu, pq RH₀¹pΩq²ˆL²0pΩq, but Remark1allows us to apply Arnold and Liu Theorem. For k1 “kand

k₂ “

"

1 if k“1, k´1 ifkě2, and l“k`1 and t“k´1, we have

}u´u_h}1,Ω0` }p´p<sub>h</sub>}0,Ω0 ďCph<sup>k</sup>` }u´u_h}´k`1,Ω1` }p´p_h}´k,Ω1q. By combining Corollary 1, Lemmas 4 and5, and inequalities (4) and (5), we get

}u´u_h}´k`1,Ω1` }p´p_h}´k,Ω1 ďCh^k h^´3ε

?2´q_ε. According to (18), withεď1,

? 1

2´qε “

?1`ε

?2ε ď 1

?ε,

therefore, taking ε“ |lnh|^´1,

}u´uh}^1,Ω0 ` }p´ph}^0,Ω0 ďCh^ka

|lnh|, which ends the proof of Theorem 1.

4 General case.

Theorem 1 and its proof have been written in the particular case of the Pk{Pk´1 finite element method, kě2, and the P1b{P¹ elements (which corresponds to the case k“1). But we can state a more general result.

First, let us focus on the assumptions: let T_h be a family of quasi-uniform simplicial triangu- lations of Ω, let V_h^k1 and W_h^k2 be two approximation spaces satisfying Assumption 2. We will also assume that V_h^k1 PCpΩq: this assumption ensures that the finite element solution is well-defined.

Moreover, we will need two more assumptions, they will play the role of Propositions 1and 2:

Assumption 3. Given B ĂΩ, consider q¹ ě2, there exists anh₀ such that for all 0ăhďh₀, we have for some positive integers k₁ and k₂:

B1r For any 1 ď ℓ, for each v P H^ℓpBq², there exists η P V_h^k1 such that, for any mesh element T ĂB,

|v´η|1,q¹,T ďCh^dp1{q1´1{2qh^r1´1|v|^ℓ,2,T, r1 “minpk1`1, ℓq.

For any 0 ď s, for each π P H^spBq, there exists ξ P W_h^k2 such that, for any mesh element T ĂB,

|π´ξ|^0,q1,T ďCh^dp1{q1´1{2qh^r2|π|^s,2,T, r₂ “minpk₂`1, sq. B3r For all vh PV_h^k1, for any mesh element T PT_h, we have

}vh}^1,q1,T ďCh^2p1{q1´1{2q}vh}^1,2,T.

AssumptionsB1r and B3r are also satisfied by a wide class of finite element spaces, including all finite element spaces defined on quasi-uniform meshes [11]. They are actually common generalisa- tions of Assumptions B1and B3.

We can now state the following result:

Theorem 2. ConsiderΩ₀ ĂĂΩ₁ ĂĂΩsatisfying Assumption 1, 1ďq ă2, letpu, pq PW₀^1,qpΩq ˆ L^q0pΩqbe the solution of Problem (1)andpuh, p_hq its Galerkin projection ontoV_h^k1ˆW_h^k2 satisfying ş

Ωp_h“0 and ż

Ω

∇pu´uhq::∇η´ ż

Ωpp´phqdivpηq “ 0 for all ηPV_h^k, ż

Ω

divpu´u_hqξ “ 0 for all ξPW_h^k.

Under the assumption that pu, pq PH^k0`1pΩ1q²ˆH^k0pΩ1q, there exists h1 such that if 0ăhďh1, we have,

}u´uh}^1,Ω0` }p´ph}^0,Ω0 ďCpΩ0,Ω1,Ωqh^k0a

|lnh|, where k0 “minpk1, k2`1q.

Proof. We will not develop the complete proof here because it is essentially the same as the proof of Theorem 1 (see Section 3). But we will explain two differences between the both proofs:

‚ the result of Lemma 1 holds in this case, but for 0ďtďminpk₁´1, k₂q.

‚ the result of Corollary 1 becomes

}u´uh}´k0`1,Ω` }p´ph}´k0,Ω

ďCh^´εh^k0p|u´u_h|1,qε,Ω` |p´p<sub>h</sub>|0,qε,Ωq, where k<sub>0</sub> “minpk<sub>1</sub>, k<sub>2</sub>`1q.

The end of the proof is the same.

5 Numerical illustrations.

In this section, we present some computations which illustrate the theoretical results proved in this paper.

Concentration of the error around the singularity. First, we define Ω as the unit square, Ω“ r0,1s².

and solve the following Stokes problem withF“^tr1,1s and x0 “ p0.5,0.5q,

$&

%

´△u`∇p “ δx₀F in Ω, divpuq “ 0 in Ω, u “ u_δ on BΩ,

(20) where u_δ is the 2d Stokeslet defined in (3).

Remark 2. Unlike Problem (1), Problem (20)has non homogeneous Dirichlet boundary conditions, but in this case, the exact solution is known: u_δ. Thus, it is easier to get some information on the error.

Figures2,3,4and 5show the repartition of the error on the velocity with aP1b{P¹ method for respectively 1{h »5,10,20 and 30. Figures 6,7,8 and 9 show the repartition of the error on the pressure for the same values of h. In both cases, they illustrate the fact that the error concentrates around the singularity. These simulations made us think that the convergence could be optimal on a subdomain which does not contain the singularity: quasi-optimality has been proved in this paper (Theorem 1).

Error 0.02

0.01

0 Figure 2: Error in velocity, 1{h»5.

Error 0.02

0.01

0 Figure 3: Error in velocity, 1{h»10.

Error 0.02

0.01

0 Figure 4: Error in velocity, 1{h»20.

Error 0.02

0.01

0 Figure 5: Error in velocity 1{h »30.

Error 1

0.75 0.5 0.25 0

Figure 6: Error in pressure, 1{h»5.

Error 1

0.75 0.5 0.25 0

Figure 7: Error in pressure, 1{h»10.

Error 1

0.75 0.5 0.25 0

Figure 8: Error in pressure, 1{h»20.

Error 1

0.75 0.5 0.25 0

Figure 9: Error in pressure 1{h»30.

Estimated orders of convergence. For this second example, the domain Ω is still the unit square, and Ω₀ is defined as the following portion of Ω,

Ω0“ txPΩ :}x´x0}²ą0.4u,

where x0 “ p0.5,0.5q. We fix F “ ^tr1,1s and solve Problem 1 for different mesh sizes h with the P1b{P1,P2{P1 and P3{P2 finite element methods.

Figure10(respectively Figure 11) presents the estimated orders of convergence for theH¹pΩ₀q- norm of the velocity (respectively the L²pΩ0q-norm of the pressure) for these three methods. The convergence far from the singularity (i.e. on Ω0) is the same as in the regular case: the Pk{Pk´1

method (or the P1b{P1 method if k “ 1) converges at the order k on Ω₀ in H¹-norm for the velocity and in L²-norm for the pressure, as proved in this paper. Let us just note that there is an over-convergence in pressure for the P1b{P¹ elements: the estimated order of convergence is approximately 2, greater than the convergence expected by Theorem 1.

About the error in L²pΩ0q-norm for the velocity, Figure 12 suggests that the Pk{Pk´1 finite element method (or P1b{P1 ifk“1) converges at the order k`1 on Ω0. This result has only been observed numerically but it is still an open question.

Elements P1b{P¹ Elements P2{P¹ Elements P3{P² Order = 1.12 Order = 2.02 Order = 3.08 1

10^´4

10^´6

10^´8 10^´2

10^´2 10^´1 0

Figure 10: Estimated order of convergence for theH¹pΩ0q-norm of the velocity.

Elements P1b{P¹ Elements P2{P1

Elements P3{P2

Order = 2.17 Order = 2.21 Order = 3.1 1

10^´4

10^´6 10^´2

10^´2 10^´1

Figure 11: Estimated order of convergence for the L²pΩ₀q-norm of the pressure.

Elements P1b{P¹ Elements P2{P1

Elements P3{P2

Order = 2.12 Order = 3.18 Order = 4.07 1

10^´3

10^´6

10^´9

10^´2 10^´1

Figure 12: Estimated order of convergence for theL²pΩ₀q-norm of the velocity.